AP Calculus BC Fundamental Theorem of Calculus PDF

Unit 6 Topic 3 The Fundamental Theorem of Calculus Name:____________________________________________ AP Calculus BC Period:______ Date:________________________________ The Fundamental Theorem of Calculus,...

Unit 6 Topic 3 The Fundamental Theorem of Calculus Name:____________________________________________ AP Calculus BC Period:______ Date:________________________________ The Fundamental Theorem of Calculus, Part 1 If f is continuous on  a, b  , then the area function a  x  b , is continuous on  a, b  and differentiable on ( a, b ). x g ( x) =  f (t )dt , for a The area function satisfies g ( x) = f ( x) ; equivalently, d x g ( x ) = f ( t ) dt = f ( x ) , which means that the area function of f is an antiderivative of f on  a, b . dx a g ( x) =  1 + sin 2 t dt , find g  ( x ) x Example 1 If 0 d 4 2 dx x Example 2 Evaluate tan t  cot t dt d 3 x5 dx  −1 Example 3 Evaluate csc t dt The Fundamental Theorem of Calculus, Part 2 If f is continuous on  a, b  and F is any antiderivative of f on  a, b  ,  f ( x ) dx = F ( b ) − F ( a ) b then a Example 4 Evaluate each definite integral.    A.  − 2  sin x dx 2 B.  − 2  cos x dx 2 C.  0 4 sec2 x dx 2 Example 5 Evaluate 0 2 x − 3 dx 3 1 x −1  1 3 4 Example 6 What is wrong with the following calculation?  dx =  = − −1 = − −1 x 2 −1  −1 3 3 Example 7 Given that f  ( x ) = 3x 2 + 4 x − 5 with the initial condition f ( 2 ) = −1 , find f ( 3) Example 8 The graph of f  on −2  x  6 consists of two line segments and a semicircle below. Given that f ( −2 ) = 5 , find f ( 0 ) , f ( 2 ) , and f ( 6 ). ** Interpretations of Definite Integrals Example 9 If p ( t ) is the rate of growth of a rabbit population, measured in rabbits per year, and there were 100 rabbits in the year 2015 ( t = 0) , write an integral expression that represents the rabbit population in 2017. Example 10 If v ( t ) is the velocity of a particle moving along the x-axis, measured in feet per second, what does  v ( t ) dt represent? Answer in correct units. 10 3 Example 11 The graph of the function f shown consists of two line segments. Let g be the function given by g ( x ) =  f ( t ) dt. x 0 A. Find g ( −1) , g  ( −1) , and g  ( −1). B. For what values of x in the open interval ( −2, 2 ) is g increasing? C. For what values of x in the open interval ( −2, 2 ) is the graph of g concave down? D. Sketch the graph of g on the closed interval  −2, 2. -3 -2 -1 1 2 3